Nonlinear growth models represent an instance of nonlinear regression models, a class of models taking the general form \[ y = \mu(x, \theta) + \epsilon, \] where \(\mu(x, \theta)\) is the mean function which depends on a possibly vector-valued parameter \(\theta\), and a possibly vector-valued predictor \(x\). The stochastic component \(\epsilon\) represents the error with mean zero and constant variance. Usually, a Gaussian distribution is also assumed for the error term.
By defining the mean function \(\mu(x, \theta)\) we may obtain several different models, all characterized by the fact that parameters \(\theta\) enter in a nonlinear way into the equation. Parameters are usually estimated by nonlinear least squares which aims at minimizing the residual sum of squares.
\[ \mu(x) = \theta_1 \exp\{\theta_2 x\} \] where \(\theta_1\) is the value at the origin (i.e. \(\mu(x=0)\)), and \(\theta_2\) represents the (constant) relative ratio of change (i.e. \(\frac{d\mu(x)}{dx }\frac{1}{\mu(x)} = \theta_2\)). Thus, the model describes an increasing (exponential growth if \(\theta_2 > 0\)) or decreasing (exponential decay if \(\theta_2 < 0\)) trend with constant relative rate.
\[ \mu(x) = \frac{\theta_1}{1+\exp\{(\theta_2 - x)/\theta_3\}} \] where \(\theta_1\) is the upper horizontal asymptote, \(\theta_2\) represents the x-value at the inflection point of the symmetric growth curve, and \(\theta_3\) represents a scale parameter (and \(1/\theta_3\) is the growth-rate parameter that controls how quickly the curve approaches the upper asymptote).
\[ \mu(x) = \theta_1 \exp\{-\theta_2 \theta_3^x\} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the value of the function at \(x = 0\) (displacement along the x-axis), and \(\theta_3\) represents a scale parameter.
The difference between the logistic and Gompertz functions is that the latter is not symmetric around the inflection point.
\[ \mu(x) = \theta_1 (1 - \exp\{-\theta_2 x\})^{\theta_3} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the rate of growth, and \(\theta_3\) in part determines the point of inflection on the y-axis.
Dipartimento della Protezione Civile: COVID-19 Italia - Monitoraggio della situazione http://arcg.is/C1unv
Source: https://github.com/pcm-dpc/COVID-19
## # Dati COVID-19 Italia
##
## ## Avvisi
##
## ```diff
## - 02/05/2020: dati Regione Lombardia ricalcolati 329 decessi (47 di oggi e 282 da riconteggio di aprile)
## - 01/05/2020: dati Regione Lazio ricalcolati 41 decessi (8 nelle ultime 48 ore e 33 ad aprile)
## - 26/04/2020: dati Regione Valle d'Aosta ricalcolati (casi testati)
## - 24/04/2020: dati Regione Sardegna ricalcolati (1.237 tamponi aggiunti)
## - 24/04/2020: dati Regione Friuli Venezia Giulia in fase di revisione su dimessi/guariti
## - 23/04/2020: dati Regione Lazio parziali (casi testati non completi)
## - 23/04/2020: dati Regione Campania parziali (casi testati non aggiornati)
## - 21/04/2020: dati Regione Lombardia parziali (casi testati non aggiornati)
## - 20/04/2020: dati Regione Lombardia ricalcolati (ricalcolo di casi testati - eliminazione duplicati)
## - 15/04/2020: dati Regione Friuli Venezia Giulia ricalcolati (ricalcolo di isolamento domiciliare e dimessi/guariti)
## - 12/04/2020: dati P.A. Bolzano ricalcolati (ricalcolo dati guariti -110 rispetto a ieri)
## - 10/04/2020: dati Regione Molise parziali (dato tamponi non aggiornato)
## - 29/03/2020: dati Regione Emilia-Romagna parziali (dato tamponi non aggiornato)
## - 26/03/2020: dati Regione Piemonte parziali (-50 deceduti - comunicazione tardiva)
## - 18/03/2020: dati Regione Campania non pervenuti
## - 18/03/2020: dati Provincia di Parma non pervenuti
## - 17/03/2020: dati Provincia di Rimini non aggiornati
## - 16/03/2020: dati P.A. Trento e Puglia non pervenuti
## - 11/03/2020: dati Regione Abruzzo non pervenuti
## - 10/03/2020: dati Regione Lombardia parziali
## - 07/03/2020: dati Brescia +300 esiti positivi
## ```
url = "https://raw.githubusercontent.com/pcm-dpc/COVID-19/master/dati-andamento-nazionale/dpc-covid19-ita-andamento-nazionale.csv"
COVID19 <- read.csv(file = url, stringsAsFactors = FALSE)
COVID19$data <- as.Date(COVID19$data)
# DT::datatable(COVID19)# create data for analysis
data = data.frame(date = COVID19$data,
y = COVID19$totale_casi,
dy = reldiff(COVID19$totale_casi))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 24831.831722 2646.674443 9.382 0.0000000000000692 ***
## th2 0.033335 0.001847 18.053 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 24720 on 68 degrees of freedom
##
## Number of iterations to convergence: 12
## Achieved convergence tolerance: 0.000007483mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 207381.5223 2330.9019 88.97 <2e-16 ***
## xmid 37.5375 0.3520 106.64 <2e-16 ***
## scal 9.0948 0.2589 35.13 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5488 on 67 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.0000009696mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
# control = nls.control(maxiter = 1000))
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 230392.2384009 1496.8435570 153.92 <2e-16 ***
## b2 7.9498889 0.1718837 46.25 <2e-16 ***
## b3 0.9397319 0.0007728 1215.95 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1940 on 67 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.0000002065richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, algorithm = "plinear",
control = nls.control(maxiter = 1000, tol = 0.1))
# algorithm is not converging...
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 239012.0711034 1588.5483012 150.46 <2e-16 ***
## th2 0.0537467 0.0007956 67.55 <2e-16 ***
## th3 5.6362411 0.1302673 43.27 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1520 on 67 degrees of freedom
##
## Number of iterations to convergence: 2
## Achieved convergence tolerance: 0.003429
# library(nlmrt)
# mod4 = nlxb(y ~ th1*(1 - exp(-th2*x))^th3,
# data = data, start = start, trace = TRUE)models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -806.3931 | 3 | 0.9045540 | 1618.786 | 1619.150 | 1625.532 | |
| Logistic model | -700.5151 | 4 | 0.9955982 | 1409.030 | 1409.646 | 1418.024 | |
| Gompertz model | -627.7086 | 4 | 0.9993685 | 1263.417 | 1264.033 | 1272.411 | |
| Richards model | -610.6516 | 4 | 0.9996110 | 1229.303 | 1229.919 | 1238.297 | *** |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Infected", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 10000),
minor_breaks = seq(0, coef(mod2)[1], by = 5000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))last_plot() +
scale_y_continuous(trans = "log10", limits = c(100,NA)) +
labs(y = "Infected (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,c("fit2", "fit3")]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Infected", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 10000),
minor_breaks = seq(0, max(ylim), by = 5000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 71 2020-05-04 264775 202195 328697
## 711 2020-05-04 202276 189387 213916
## 712 2020-05-04 209239 204230 213686
## 713 2020-05-04 210828 206988 214645
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Infected", color = "Model") +
scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 10000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# create data for analysis
data = data.frame(date = COVID19$data,
y = COVID19$deceduti,
dy = reldiff(COVID19$deceduti))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 2597.60063 305.20935 8.511 0.00000000000259 ***
## th2 0.03719 0.00200 18.596 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3238 on 68 degrees of freedom
##
## Number of iterations to convergence: 13
## Achieved convergence tolerance: 0.000005172mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 28511.2915 334.1133 85.33 <2e-16 ***
## xmid 40.5589 0.3399 119.31 <2e-16 ***
## scal 8.6200 0.2437 35.37 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 717.6 on 67 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000002056mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# manually set starting values
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
# control = nls.control(maxiter = 10000))
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 32054.9066062 216.8629273 147.81 <2e-16 ***
## b2 10.6475670 0.2592695 41.07 <2e-16 ***
## b3 0.9377425 0.0007845 1195.37 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 245.1 on 67 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000002556richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, algorithm = "port",
control = nls.control(maxiter = 1000))
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 32988.3955728 224.7449978 146.78 <2e-16 ***
## th2 0.0578124 0.0008169 70.77 <2e-16 ***
## th3 8.0581024 0.2064624 39.03 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 201.1 on 67 degrees of freedom
##
## Number of iterations to convergence: 4
## Achieved convergence tolerance: 0.000005386models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -664.1033 | 3 | 0.9160388 | 1334.2067 | 1334.5703 | 1340.9522 | |
| Logistic model | -558.1057 | 4 | 0.9961178 | 1124.2114 | 1124.8268 | 1133.2054 | |
| Gompertz model | -482.9145 | 4 | 0.9994751 | 973.8289 | 974.4443 | 982.8229 | |
| Richards model | -469.0555 | 4 | 0.9996412 | 946.1110 | 946.7264 | 955.1050 | *** |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Deceased", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))last_plot() +
scale_y_continuous(trans = "log10", limits = c(10,NA)) +
labs(y = "Deceased (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Deceased", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 71 2020-05-04 36412 28630 44847
## 711 2020-05-04 27701 26015 28974
## 712 2020-05-04 28688 28118 29263
## 713 2020-05-04 28850 28341 29322
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Deceased", color = "Model") +
scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# create data for analysis
data = data.frame(date = COVID19$data,
y = COVID19$dimessi_guariti,
dy = reldiff(COVID19$dimessi_guariti))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 2313.368701 183.326828 12.62 <2e-16 ***
## th2 0.052399 0.001284 40.80 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3455 on 68 degrees of freedom
##
## Number of iterations to convergence: 10
## Achieved convergence tolerance: 0.000004975mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 113693.5328 3594.7930 31.63 <2e-16 ***
## xmid 59.5106 0.8096 73.51 <2e-16 ***
## scal 11.6730 0.2703 43.19 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1242 on 67 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000003055mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 258204.6365469 13495.9431136 19.13 <2e-16 ***
## b2 7.9151456 0.1087423 72.79 <2e-16 ***
## b3 0.9727367 0.0007706 1262.39 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 667.6 on 67 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000003334richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, # algorithm = "port",
control = nls.control(maxiter = 1000))
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 775008.991501 137423.288482 5.640 0.0000003710797866 ***
## th2 0.010888 0.001138 9.565 0.0000000000000376 ***
## th3 3.555525 0.117451 30.272 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 553.1 on 67 degrees of freedom
##
## Number of iterations to convergence: 29
## Achieved convergence tolerance: 0.000003742models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -668.6443 | 3 | 0.9855230 | 1343.289 | 1343.652 | 1350.034 | |
| Logistic model | -596.4793 | 4 | 0.9980360 | 1200.959 | 1201.574 | 1209.953 | |
| Gompertz model | -553.0496 | 4 | 0.9993580 | 1114.099 | 1114.715 | 1123.093 | |
| Richards model | -539.8784 | 4 | 0.9995383 | 1087.757 | 1088.372 | 1096.751 | *** |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Recovered", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))last_plot() +
scale_y_continuous(trans = "log10", limits = c(10,NA)) +
labs(y = "Recovered (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Recovered", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 71 2020-05-04 95489 85789 104837
## 711 2020-05-04 82764 78640 85909
## 712 2020-05-04 84919 82919 86464
## 713 2020-05-04 85756 84498 86958
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Recovered", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 5000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))df = data.frame(date = COVID19$data,
positives = c(NA, diff(COVID19$totale_casi)),
swabs = c(NA, diff(COVID19$tamponi)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
# df$y = df$positives/df$swabs
df$y = df$positives/c(NA, zoo::rollmean(df$swabs, 2))
df = subset(df, swabs > 50)
# DT::datatable(df[,-4], )ggplot(df, aes(x = date)) +
geom_point(aes(y = swabs, color = "swabs"), pch = 19) +
geom_line(aes(y = swabs, color = "swabs")) +
geom_point(aes(y = positives, color = "positives"), pch = 0) +
geom_line(aes(y = positives, color = "positives")) +
labs(x = "", y = "Number of cases", color = "") +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = palette()[c(2,1)]) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))ggplot(df, aes(x = date, y = y)) +
geom_smooth(method = "loess", se = TRUE, col = "black") +
geom_point(col=palette()[4]) +
geom_line(size = 0.5, col=palette()[4]) +
labs(x = "", y = "% positives among admnistered swabs (two-day rolling mean)") +
scale_y_continuous(labels = scales::percent_format(),
breaks = seq(0, 0.5, by = 0.05)) +
coord_cartesian(ylim = c(0,max(df$y, na.rm = TRUE))) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))df = data.frame(date = COVID19$data,
hospital = c(NA, diff(COVID19$totale_ospedalizzati)),
icu = c(NA, diff(COVID19$terapia_intensiva)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1ggplot(df, aes(x = date, y = hospital)) +
geom_smooth(method = "loess", se = TRUE, col = "black") +
geom_point(col = "orange") +
geom_line(size = 0.5, col = "orange") +
labs(x = "", y = "Change hospitalized patients") +
coord_cartesian(ylim = range(df$hospital, na.rm = TRUE)) +
scale_y_continuous(minor_breaks = seq(min(df$hospital, na.rm = TRUE),
max(df$hospital, na.rm = TRUE),
by = 100)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))ggplot(df, aes(x = date, y = icu)) +
geom_smooth(method = "loess", se = TRUE, col = "black") +
geom_point(col = "red2") +
geom_line(size = 0.5, col = "red2") +
labs(x = "", y = "Change ICU patients") +
coord_cartesian(ylim = range(df$icu, na.rm = TRUE)) +
scale_y_continuous(minor_breaks = seq(min(df$icu, na.rm = TRUE),
max(df$icu, na.rm = TRUE),
by = 10)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))